Adaptive Control and the Definition of Exponential Stability Travis - PowerPoint PPT Presentation

Adaptive Control and the Definition of Exponential Stability Travis E. Gibson † and Anuradha M. Annaswamy ‡ † ‡ American Control Conference, Chicago IL July 1, 2015

Objectives Prove that the following statement is incorrect ◮ “If the reference model is persistently exciting then the adaptive system is globally exponentially stable” 2 / 15

Objectives Prove that the following statement is incorrect ◮ “If the reference model is persistently exciting then the adaptive system is globally exponentially stable” Prove the following ◮ Adaptive systems can at best be uniformly asymptotically stable in the large 2 / 15

Objectives Prove that the following statement is incorrect ◮ “If the reference model is persistently exciting then the adaptive system is globally exponentially stable” Prove the following ◮ Adaptive systems can at best be uniformly asymptotically stable in the large Main insights ◮ Indeed if the reference model is PE then after some time the plant will be PE, but after exactly how much time ? ◮ We will show how a PE condition on the reference model implies a weak PE condition on the plant state. 2 / 15

Outline ◮ Definitions ◮ Stability ◮ Exponential Stability ◮ Persistent Excitation (PE) ◮ weak Persistent Excitation (PE ∗ ) ◮ Link between PE and Exponential Stability ◮ Link between PE ∗ and Uniform Asymptotic Stability ◮ Simulation Studies 3 / 15

Uniform Stability in the Large (Global) x ( t ) = f ( x ( t ) , t ) ˙ x 0 � x ( t 0 ) s Solution s ( t ; x 0 , t 0 ) x 0 t 4 / 15

Uniform Stability in the Large (Global) x ( t ) = f ( x ( t ) , t ) ˙ ǫ x 0 � x ( t 0 ) s δ Solution s ( t ; x 0 , t 0 ) x 0 t t 0 Definition: Uniform Stability in the Large (Massera, 1956) (i) Uniformly Stable : ∀ ǫ > 0 ∃ δ ( ǫ ) > 0 s.t. � x 0 � ≤ δ = ⇒ � s ( t ; x 0 , t 0 ) � ≤ ǫ . 4 / 15

Uniform Stability in the Large (Global) η x ( t ) = f ( x ( t ) , t ) ˙ ǫ x 0 � x ( t 0 ) s t 0 + T δ, ρ Solution s ( t ; x 0 , t 0 ) x 0 t t 0 Definition: Uniform Stability in the Large (Massera, 1956) (i) Uniformly Stable : ∀ ǫ > 0 ∃ δ ( ǫ ) > 0 s.t. � x 0 � ≤ δ = ⇒ � s ( t ; x 0 , t 0 ) � ≤ ǫ . (ii) Uniformly Attracting in the Large : For all ρ , η ∃ T ( η , ρ ) � x 0 � ≤ ρ = ⇒ � s ( t ; x 0 , t 0 ) � ≤ η ∀ t ≥ t 0 + T . 4 / 15

Uniform Stability in the Large (Global) η x ( t ) = f ( x ( t ) , t ) ˙ ǫ x 0 � x ( t 0 ) s t 0 + T δ, ρ Solution s ( t ; x 0 , t 0 ) x 0 t t 0 Definition: Uniform Stability in the Large (Massera, 1956) (i) Uniformly Stable : ∀ ǫ > 0 ∃ δ ( ǫ ) > 0 s.t. � x 0 � ≤ δ = ⇒ � s ( t ; x 0 , t 0 ) � ≤ ǫ . (ii) Uniformly Attracting in the Large : For all ρ , η ∃ T ( η , ρ ) � x 0 � ≤ ρ = ⇒ � s ( t ; x 0 , t 0 ) � ≤ η ∀ t ≥ t 0 + T . (iii) Uniformly Asymptotically Stable in the Large (UASL) = uniformly stable + uniformly bounded + uniformly attracting in the large . 4 / 15

Exponential Stability κ � x 0 � x ( t ) = f ( x ( t ) , t ) ˙ κ � x 0 � e − ν ( t − t 0 ) x 0 x 0 � x ( t 0 ) ρ s Solution s ( t ; x 0 , t 0 ) t 0 t Definition: (Malkin, 1935; Kalman and Bertram, 1960) (i) Exponentially Stable (ES): ∀ ρ > 0 ∃ ν ( ρ ) , κ ( ρ ) s.t. ⇒ � s ( t ; x 0 , t 0 ) � ≤ κ � x 0 � e − ν ( t − t 0 ) � x 0 � ≤ ρ = 5 / 15

Exponential Stability κ � x 0 � x ( t ) = f ( x ( t ) , t ) ˙ κ � x 0 � e − ν ( t − t 0 ) x 0 x 0 � x ( t 0 ) ρ s Solution s ( t ; x 0 , t 0 ) t 0 t Definition: (Malkin, 1935; Kalman and Bertram, 1960) (i) Exponentially Stable (ES): ∀ ρ > 0 ∃ ν ( ρ ) , κ ( ρ ) s.t. ⇒ � s ( t ; x 0 , t 0 ) � ≤ κ � x 0 � e − ν ( t − t 0 ) � x 0 � ≤ ρ = (ii) Exponentially Stable in the Large (ESL): ∃ ν , κ s.t. � s ( t ; x 0 , t 0 ) � ≤ κ � x 0 � e − ν ( t − t 0 ) 5 / 15

Persistent Excitation ω : [ t 0 , ∞ ) → R p “Exogenous Signal” : Initial Condition : ω 0 = ω ( t 0 ) y ( t, ω ) : [ t 0 , ∞ ) × R p → R m Parameterized Function : 6 / 15

Persistent Excitation ω : [ t 0 , ∞ ) → R p “Exogenous Signal” : Initial Condition : ω 0 = ω ( t 0 ) y ( t, ω ) : [ t 0 , ∞ ) × R p → R m Parameterized Function : Definition (i) Persistently Exciting (PE) : ∃ T , α s.t. � t + T y ( τ, ω ) y T ( τ, ω ) dτ ≥ α I t for all t ≥ t 0 and ω 0 ∈ R p . 6 / 15

Persistent Excitation ω : [ t 0 , ∞ ) → R p “Exogenous Signal” : Initial Condition : ω 0 = ω ( t 0 ) y ( t, ω ) : [ t 0 , ∞ ) × R p → R m Parameterized Function : Definition (i) Persistently Exciting (PE) : ∃ T , α s.t. � t + T y ( τ, ω ) y T ( τ, ω ) dτ ≥ α I t for all t ≥ t 0 and ω 0 ∈ R p . (ii) weakly Persistently Exciting (PE ∗ ( ω, Ω)) : ∃ a compact set Ω ⊂ R p , T (Ω) > 0 , α (Ω) s.t. � t + T y ( τ, ω ) y T ( τ, ω ) dτ ≥ α I t for all ω 0 ∈ Ω and t ≥ t 0 . 6 / 15

properties of adaptive control 7 / 15

Adaptive Control x m Reference Model - x = Ax − Bθ T x + Bu Plant ˙ e � r Reference Model x m = Ax m + Br ˙ x � Plant ˆ θ T 8 / 15

Adaptive Control x m Reference Model - x = Ax − Bθ T x + Bu Plant ˙ e � r Reference Model x m = Ax m + Br ˙ x � Plant ˆ θ T Unknown Parameter θ 8 / 15

Adaptive Control x m Reference Model - x = Ax − Bθ T x + Bu Plant ˙ e � r Reference Model x m = Ax m + Br ˙ x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) 8 / 15

Adaptive Control x m Reference Model - x = Ax − Bθ T x + Bu Plant ˙ e � Reference Model x m = Ax m + Br ˙ r x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Error e = x − x m θ ( t ) = ˆ ˜ Parameter Error θ ( t ) − θ Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) 8 / 15

Adaptive Control x m Reference Model x = Ax − Bθ T x + Bu - Plant ˙ e � Reference Model x m = Ax m + Br ˙ r x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Error e = x − x m θ ( t ) = ˆ ˜ Parameter Error θ ( t ) − θ Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) ˙ ˆ θ ( t ) = − xe T PB Update Law 8 / 15

Adaptive Control x m Reference Model x = Ax − Bθ T x + Bu - Plant ˙ e � Reference Model x m = Ax m + Br ˙ r x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Error e = x − x m θ ( t ) = ˆ ˜ Parameter Error θ ( t ) − θ Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) ˙ ˆ θ ( t ) = − xe T PB Update Law � � V ( e ( t ) , ˜ ˜ θ T ( t )˜ θ ( t )) = e T ( t ) Pe ( t ) + Trace Stability θ ( t ) 8 / 15

Adaptive Control x m Reference Model x = Ax − Bθ T x + Bu - Plant ˙ e � Reference Model x m = Ax m + Br ˙ r x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Error e = x − x m θ ( t ) = ˆ ˜ Parameter Error θ ( t ) − θ Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) ˙ ˆ θ ( t ) = − xe T PB Update Law � � V ( e ( t ) , ˜ ˜ θ T ( t )˜ θ ( t )) = e T ( t ) Pe ( t ) + Trace Stability θ ( t ) V ≤ e T Qe ˙ 8 / 15

Adaptive Control x m Reference Model x = Ax − Bθ T x + Bu - Plant ˙ e � Reference Model x m = Ax m + Br ˙ r x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Error e = x − x m θ ( t ) = ˆ ˜ Parameter Error θ ( t ) − θ Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) ˙ ˆ θ ( t ) = − xe T PB Update Law � � V ( e ( t ) , ˜ ˜ θ T ( t )˜ θ ( t )) = e T ( t ) Pe ( t ) + Trace Stability θ ( t ) V ≤ e T Qe ˙ � V ( e ( t 0 ) , ˜ � e � L ∞ ≤ θ ( t 0 )) /P min � V ( e ( t 0 ) , ˜ � e � L 2 ≤ θ ( t 0 )) /Q min 8 / 15

Adaptive Control x m Reference Model x = Ax − Bθ T x + Bu - Plant ˙ e � Reference Model x m = Ax m + Br ˙ r x � Plant u = ˆ θ T ( t ) x + r Control Input ˆ θ T Error e = x − x m θ ( t ) = ˆ ˜ Parameter Error θ ( t ) − θ Unknown Parameter θ ˆ Adaptive Parameter θ ( t ) ˙ ˆ θ ( t ) = − xe T PB Update Law � � V ( e ( t ) , ˜ ˜ θ T ( t )˜ θ ( t )) = e T ( t ) Pe ( t ) + Trace Stability θ ( t ) V ≤ e T Qe ˙ � V ( e ( t 0 ) , ˜ � e � L ∞ ≤ θ ( t 0 )) /P min � V ( e ( t 0 ) , ˜ � e � L 2 ≤ θ ( t 0 )) /Q min The L-norms of e are initial condition dependent!! 8 / 15

Exponential Stability and Adaptive Control Bx T ( t ) � � � e ( t ) � A z ( t ) � z ( t ) = ˙ z ( t ) , − x ( t ) B T P ˜ 0 θ ( t ) 9 / 15

Exponential Stability and Adaptive Control Bx T ( t ) � � � e ( t ) � A z ( t ) � z ( t ) = ˙ z ( t ) , − x ( t ) B T P ˜ 0 θ ( t ) Theorem: (Morgan and Narendra, 1977) If x ( t ) ∈ PE then z ( t ) = 0 is UASL. 9 / 15

Exponential Stability and Adaptive Control Bx T ( t ) � � � e ( t ) � A z ( t ) � z ( t ) = ˙ z ( t ) , − x ( t ) B T P ˜ 0 θ ( t ) Theorem: (Morgan and Narendra, 1977) If x ( t ) ∈ PE then z ( t ) = 0 is UASL. ESL ES UASL UAS + Linear 9 / 15